We note that Dirichlet boundary conditions can be included as data in \(\lbrace u_i,x_i\rbrace \) . This is the basic structure of physics-informed neural networks that can be accommodated for a large class of problems [1]}. One issue of this formulation is that the neural network that represents the solution is parametrized with a vector \(\mbox{$x$}{} \in \mathbb {R}^3\) , which is a Euclidean space. In this way, it is difficult to inform the network about the geometry of the domain (in case of a problem formulated on a manifold) or, in the case of a complex domain in \(\mathbb {R}^d\)  where the boundaries lie. This is important in the computation of the differential operators. Furthermore, points in the manifold \(\mathcal {B}\)  that are close in the Euclidean space are not necessarily close in the intrinsic distance, as illustrated in Figure REF .
